# [seqfan] Occupancy Perfect Squares - more cases

Ron Hardin rhhardin at att.net
Thu Dec 20 18:33:07 CET 2012

```Surveying the queued projects for perfect squares in even T(n,k) rows or columns
I find many cases.

cuf and cvh are the two mentioned here already.

Perhaps there is a revealing common thread among them.

=== all perfect squares in even cols ===

cue
T(n,k)=Number of nXk arrays of occupancy after each element moves to some
horizontal or antidiagonal neighbor

Table starts
_0____1______2________4________8______16_______32_____64_128
_0____4_____24______169_____1155____7921____54288_372100____
_0___16____270_____6889___154440_3538161_80655102___________
_0___64___3024___279841_20489472____________________________
_0__256__33858_11363641_____________________________________
_0_1024_379080______________________________________________
_0_4096_____________________________________________________
_0__________________________________________________________

=== all perfect squares in even cols ===

cuf
T(n,k)=Number of nXk arrays of occupancy after each element moves to some
horizontal, diagonal or antidiagonal neighbor

Table starts
_0_______1__________2_____________4______________8_____________16
_0_______9_________50___________484___________3864__________32400
_0______64_______1056_________45369________1337130_______44315649
_0_____441______21660_______4096576______438377904____56606878084
_0____3025_____441720_____367565584___142283547000_71306050492681
_0___20736____8997120___32940435025_46087307409528_______________
_0__142129__183206016_2951293792489______________________________
_0__974169_3730330800____________________________________________
_0_6677056_______________________________________________________
_0_______________________________________________________________

=== all perfect squares in even cols or rows ===

cus
T(n,k)=Number of nXk arrays of occupancy after each element moves to some
horizontal or vertical neighbor, with no occupancy greater than 2

Table starts
___0______1________2__________4__________8_________16_______32________64
___1______9_______49________289_______1681_______9801____57121____332929
___2_____49______656______12544_____200072____3485689_58132316_990486784
___4____289____12544_____643204___31203396_1539699121___________________
___8___1681___200072___31203396_4235650596______________________________
__16___9801__3485689_1539699121_________________________________________
__32__57121_58132316____________________________________________________
__64_332929_____________________________________________________________
_128____________________________________________________________________

=== all perfect squares in even cols ===

cuv
T(n,k)=Number of nXk arrays of occupancy after each element moves to some
horizontal, diagonal or antidiagonal neighbor, with no occupancy greater than 2

Table starts
_0_____1____2________4_______8_________16_______32_____64_128
_0_____9___10______196_____720_______6400____34272_242064____
_0____49___46_____9025___58700____2518569_32085376___________
_0___289__212___427716_4984812_1026177156____________________
_0__1681__976_20277009_______________________________________
_0__9801_4492________________________________________________
_0_57121_____________________________________________________
_0___________________________________________________________

=== all perfect squares in even cols ===

cuw
T(n,k)=Number of nXk arrays of occupancy after each element moves to some
horizontal or antidiagonal neighbor, with no occupancy greater than 2

Table starts
_0_____1_____2__________4__________8__________16_________32________64____128
_0_____4_____8_________81________392________2500______14112_____83521_484128
_0____16____30_______2116______22268______488601____7356420_132273001_______
_0____64___112______58081____1319409___103489929_4085800416_________________
_0___256___418____1612900___78633756_22466712321____________________________
_0__1024__1560___44903401_4684821704________________________________________
_0__4096__5822_1250753956___________________________________________________
_0_16384_21728______________________________________________________________
_0_65536____________________________________________________________________
_0__________________________________________________________________________

=== all perfect squares in even cols or rows ===

cvh
T(n,k)=Number of nXk arrays of occupancy after each element moves to some
horizontal or vertical neighbor, with every occupancy equal to zero or two

Table starts
_0___0_____0______1____0_____0______0______1_0____0_0
_0___4_____0_____16____0____64______0____256_0_1024__
_0___0_____0_____81____0_____0______0__11025_0_______
_1__16____81____529_3249_20164_124609_770884_________
_0___0_____0___3249____0_____0_______________________
_0__64_____0__20164____0_____________________________
_0___0_____0_124609__________________________________
_1_256_11025_________________________________________
_0___0_______________________________________________
_0___________________________________________________

rhhardin at mindspring.com
rhhardin at att.net (either)

```